Published by : International Journal of Engineering Research & Technology (IJERT) http://www.ijert.org ISSN: 2278-0181 Vol. 8 Issue 05, May-2019 Extensive Study of 2d Transformations in Computer Graphics Ankur Rana Sandhya Samant Assistant Professor, Assistant Professor, Department of Computer Science Department of Computer Science Quantum University, Roorkee, India Quantum University, Roorkee, India Abhishek Agarwal Assistant Professor, Department of Mathematics Quantum University, Roorkee , India Transformation Abstract:- This research paper gives a comprehensive overview of various 2D Transformations in Computer Graphics. 2D graphics uses a two dimensional representation of the real world objects, stored as images in the computer for Translation Scaling Roatation Reflection Shearing being manipulated and rendered. Transformation, in graphics, is the process of manipulation of images. Majorly, it is used to reposition the image (Translation), to change its size Figure 1 Types of Transformation (Scaling) or orientation (Rotation) and also to get the mirror image (Reflection). 1. Translation: Translation is the transformation that moves Keywords: 2D Graphics, Objects, Images, Transformation, every point by certain distance in horizontal and vertical Translation, Scaling, Rotation, Reflection etc. directions[2]. This movement is based on translation factors tx and ty where, tx is the horizontal movement and I. INTRODUCTION ty is the vertical movement. Computer graphics is the field of computer science which Let the point P be defined by (x,y) coordinates, then the deals with creation, storing, displaying and manipulation of new coordinates (x’,y’) after translation[3] will be: images. It is concerned with digitally synthesizing and manipulating visual contents. An image is the 2D x’ = x+ tx representation of a 3D object. Images, on screen, are made y’ = y+ ty up of pixels, which is the smallest addressable unit of display. Computer graphics is generally categorized into Interactive and Non-interactive graphics. Interactive Computer Graphics is a two way communication between the computer and user, where the image on the screen is controlled by giving signals or instructions to the computer through an input device, Whereas Non-Interactive Computer Graphics is a passive system in which the user does not have any command or control over the image. Here the image is a product of already stored program and is operated by the instructions according to the program and not the user. Figure 2 Translation Example of Interactive system is Flight Simulator and the example of Non-Interactive system is screen savers. Figure 2 shows the translation applied on alphabet F. II . TRANSFORMATION Furthermore, the equation of translation can be explained using matrix, where the representation of the operation will Transformation is the process of changing the original be: coordinates of the image with the new coordinates by x′ 1 0 tx x applying different mathematical functions.[1] [y′] = [0 1 ty].[y] 1 0 0 1 1 Figure 3 Translation Matrix IJERTV8IS050398 www.ijert.org 621 (This work is licensed under a Creative Commons Attribution 4.0 International License.) Published by : International Journal of Engineering Research & Technology (IJERT) http://www.ijert.org ISSN: 2278-0181 Vol. 8 Issue 05, May-2019 2. Scaling: Scaling is the transformation in which we resize the object. It refers to enlargement or shrinking of Figure 6 shows the operation of Rotation. the object. Scaling depends on the scaling factors Sx and Sy where Sx and Sy are the scaling in horizontal and Furthermore, we can also represent the rotation on ʘ angle vertical directions respectively. with the given matrix: Let the point P be defined by (x,y) coordinates, then the new coordinates (x’,y’) after scaling will be: x′ cosʘ −sinʘ 0 x x’ = x . Sx [y′]=[sinʘ cosʘ 0].[y] y’ = y . Sy 1 0 0 1 1 Figure 7 Rotation Matrix 4. Reflection: Reflection is the transformation that produces the mirror image of an object. The mirror image of the object is produced with respect to the reflection axis by rotating the object at 180 degrees along the reflection axis. The axis of reflection can be anywhere in the x-y plane. If x-axis is the axis of reflection then the new coordinates after reflection will be: X’ = X and Y’ = -Y, If y-axis is the axis of reflection then the new coordinates after reflection will be: X’ = -X and Y’ = Y, Figure 4 Scaling If both x-axis and y-axis are the axis of reflection then the new coordinates will be: Figure 3 shows the scaling applied on alphabet F. X’ = -X and Y’ = -Y. Scaling can be categorized into uniform and non-uniform scaling. If Sx = Sy then it is Uniform Scaling and if Sx ≠ Sy then it is Non-uniform Scaling. Furthermore, the matrix representation of the above operation can be given as under: x′ sx 0 0 x [y′]=[ 0 sy 0].[y] 1 0 0 1 1 Figure 5 Scaling Matrix 3. Rotation: Rotation is the transformation in which we Figure 8 Reflection rotate each point through the origin (0,0). Every point is rotated through the same angle, which is known as the Figure 8 shows the reflection when both x-axis and y-axis angle of rotation[4]. The unit of this angle can either be in are the axis of reflection. degrees or in radians. If the angle of rotation is ʘ then it is x′ 1 0 0 x taken positive for counter clockwise direction and negative [y′]=[0 −1 0].[y] for clockwise direction. The equation for the new 1 0 0 1 1 coordinates of the point after rotation is: Figure 9 Reflection Matrix (about x axis) X’ = x cos(ʘ) – y sin (ʘ) x′ −1 0 0 x Y’ = y cos (ʘ) + x sin (ʘ) [y′]=[ 0 1 0].[y] 1 0 0 0 1 Figure 10 Reflection Matrix (about y axis) x′ −1 0 0 x [y′]=[ 0 −1 0].[y] 1 0 0 1 1 Figure 11 Reflection Matrix (about xy axis) 5. Shearing: Shearing is the transformation that produces the distortion of the object. In other words we can also say that shearing tilts the object. If the shearing is horizontal then it will tilt the object towards the left (negative shear) or right (positive shear). Whereas the vertical shear tilts the object up or down. Figure 6 Rotation IJERTV8IS050398 www.ijert.org 622 (This work is licensed under a Creative Commons Attribution 4.0 International License.) Published by : International Journal of Engineering Research & Technology (IJERT) http://www.ijert.org ISSN: 2278-0181 Vol. 8 Issue 05, May-2019 Figure 11 Shearing Figure 9 shows an example of horizontal shear. The equation for horizontal shear applied to a point (x,y), to obtain new coordinates (X’,Y’) is given by: X’ = x + b.y and Y’ = y, Where b is shearing factor. Whereas the equation for vertical shear with shearing factor c is given by: X’ = x and Y’ = c . x + y The matrix for X – shear can be represented as: 1 0 0 Xsh=[shx 1 0] 0 0 1 Figure 12 Horizontal Shearing Similarly, the matrix for vertical shearing can be represented as: 1 shy 0 Ysh=[0 1 0] 0 0 1 Figure 13 Vertical Shearing III . CONCLUSION Here in this paper we have discussed various transformations along with their matrix representations. Among the transformation we have discussed, Translation, Rotation, Reflection and their combinations are the rigid transformations because in these transformations the pre-image and image are of same size and shape (congruent). Scaling and Shearing are not rigid transformations because they produce images with different size or shape. IV. REFERENCES [1] Ms.A.J.Rajeswari Joe, Dr.N.Rama, “Scaling Transform Methods Advanced Computing : An International Journal ( ACIJ ), Vol.4, No.2, March 2013 [2] David J.EckHobart and William Smith, “Introduction to Computer Graphics Version 1.2”, January 2018 [3] Kartik, Seth Jai Parkash Mukand Lal. “A Descriptive Study of 2D Graphics Transformations”, ISSN(Online): 2320-9801ISSN (Print): 2320-9798 International Journal of Innovative Research in Computer and Communication Engineering [4] Manoj Kumar Srivastav Champdani Adarsh Sharmik Vidyamandir, 3, R. B. S. Road, Champdani, “transformation of an Object in Computer Graphics: A Case Study of Mathematical Matrix Theory “(Elixir International Journal IJERTV8IS050398 www.ijert.org 623 (This work is licensed under a Creative Commons Attribution 4.0 International License.).